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        <title>Tangent Space of Manifolds</title>

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                <h1 id="title" titleSize="">
                    Tangent Space of Manifolds
                </h1>
            
            <h1 id="motivation--definition">Motivation &amp; Definition</h1>
<p>In elementary calculus on $\R^2$, one uses the techniques of derivatives, and hence <em>tangent lines at a point</em>, to study the local behaviour of curves. This philosophy of probing the local structure of the space via derivatives is the <em>raison d&rsquo;être</em> of <a href=https://zhaoshenzhai.github.io/mathwiki/smooth_manifold.md class="internalLink references" title="Smooth Manifold" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/smooth_manifold.md&#34;, {&#34;Date&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Smooth Manifold&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/smooth_manifold&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Smooth Manifold&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/smooth_manifold&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/smooth_manifold.md&#34;, {&#34;Date&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Smooth Manifold&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/smooth_manifold&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">smooth</a> <a href=https://zhaoshenzhai.github.io/mathwiki/manifold.md class="internalLink references" title="Manifold" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/manifold.md&#34;, 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onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-08-28T19:20:34-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-28T19:20:34-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-28T19:20:34-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Manifold&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/manifold&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/manifold.md&#34;, 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<br>
<p>  However, our definition must be intrinsic to the smooth structure of a manifold $M$ and not depend on any embedding $M\into\R^k$, so a certain level of abstraction is needed. The key observation is that the ‘space of directions’ at a point can be identified with the set of directional derivatives at that point, so we take those <a href=https://zhaoshenzhai.github.io/mathwiki/derivation.md class="internalLink references ghostLink" title="derivations" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/derivation.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/derivation.md&#34;, &#34;nopPage&#34;);">derivations</a> as our starting point. To emphasize that the tangent space at a point $p\in M$ depends only on the <em>local</em> structure around $p$, we shall descend to the level of <a href=https://zhaoshenzhai.github.io/mathwiki/smooth_manifold.md/#sheaf_of_smooth_functions class="internalLink references" title="Smooth Manifold" mathLink="" secID="sheaf_of_smooth_functions" secDisplay="Sheaf of Smooth Functions" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/smooth_manifold.md/#sheaf_of_smooth_functions&#34;, {&#34;Date&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Smooth Manifold&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/smooth_manifold&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Smooth Manifold&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/smooth_manifold&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/smooth_manifold.md/#sheaf_of_smooth_functions&#34;, {&#34;Date&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-04T20:48:47-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Smooth Manifold&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/smooth_manifold&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">germs</a>, denoted $C^\infty_p(M)$:</p>
<div class="env envDef" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/definition.svg"><b class="envTitle">Definition. </b>Let $M$ be a smooth manifold and $p\in M$. The <em>tangent space</em> of $M$ at $p$ is the <a href=https://zhaoshenzhai.github.io/mathwiki/vector_space.md class="internalLink references" title="Vector Space" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/vector_space.md&#34;, {&#34;Date&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Vector Space&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/vector_space&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Vector Space&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/vector_space&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/vector_space.md&#34;, {&#34;Date&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Vector Space&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/vector_space&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">vector space</a> of all derivations at $p$:
$$\begin{equation}
T_pM\coloneqq\l\{v\in\Hom_\R(C_p^\infty(M),\R)\st v(fg)=v(f)g(p)+f(p)v(g)\r\}.
\end{equation}$$</div>

<h2 id="the-differential">The Differential</h2>
<p>Every smooth map $f:M\to N$ induces a linear map $T_pf:T_pM\to T_{f(p)}N$ between tangent spaces by push-forward of derivations, i.e., $v\mapsto(g\mapsto v(f\circ g))$. Since $T_p(g\circ f)=T_{f(p)}g\circ T_pf$, this defines a <a href=https://zhaoshenzhai.github.io/mathwiki/functor.md class="internalLink references" title="Functor" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/functor.md&#34;, 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onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-05-28T16:05:23-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-28T16:05:23-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-28T16:05:23-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Functor&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/functor&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/functor.md&#34;, {&#34;Date&#34;:&#34;2024-05-28T16:05:23-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-28T16:05:23-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-28T16:05:23-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Functor&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/functor&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">functor</a> $T:\catman_\blob\to\catvect[\R]$ from the <a href=https://zhaoshenzhai.github.io/mathwiki/category.md class="internalLink references" title="Category" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/category.md&#34;, {&#34;Date&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Category&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/category&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Category&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/category&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/category.md&#34;, {&#34;Date&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Category&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/category&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">category</a> of pointed manifolds, which informally forgets everything but the local properties of manifolds at a fixed point.</p>
<br>
<p>  Let us apply this functor to any chart $(U,\phi)$ for $M$ centered at $p$. Note that $C_p^\infty(U)=C_p^\infty(M)$, so the tangent spaces $T_pU,T_pM$ coincide, and hence $T_p\phi:T_pM\to T_{\widehat{p}}\R^n$. This is an isomorphism, since $\phi$ is. We claim that $\dim T_\widehat{p}\R^n=n$, so $\dim T_pM=n$ too.</p>
<div class="space"></div>
<div class="collapsibleContainer" id=""><i class="proofHeader collapsibleHeaderButton collapsibleHeader noSelect">Proof.</i><span class="collapsibleHintText noSelect"><i> Click to expand...</i></span>

        <span class="collapsibleContent">We claim that for every $p\in\R^n$, the map $\R^n\to T_p\R^n$ sending $v\mapsto v^i\del_i|_p$ is an isomorphism. This map is clearly linear, and is injective since if $v^i\del_i|_p=0$, then $0=v^i\del_i|_p(x^j)=v^j$ for all $j$, and hence $v=0$. To show that it is surjective, let $D\in T_p\R^n$ be a derivation. For each $j$, let $v^j\coloneqq D(x^j)$; we claim that $D=v^i\del_i|_p$. Indeed, for every $f\in C^\infty(\R^n)$, we have<a href=https://zhaoshenzhai.github.io/mathwiki/calculus_on_R.md/#fundamental_theorem_of_calculus class="internalLink proved_by ghostLink dag" title="" mathLink="" secID="fundamental_theorem_of_calculus" secDisplay="$^\dagger$" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/calculus_on_R.md/#fundamental_theorem_of_calculus&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/calculus_on_R.md/#fundamental_theorem_of_calculus&#34;, &#34;nopPage&#34;);">$^\dagger$</a>
$$\begin{equation}
f(x)-f(p)=\int_0^1\frac{\d}{\d t}f(p+t(x-p))\,\d t=(x^i-p^i)\int_0^1\del_if(p+t(x-p))\,\d t\eqqcolon(x^i-p^i)h_i(x),
\end{equation}$$
so using that $D(\textrm{const})=0$, we obtain $Df=D(x^i)h_i(p)$. But $D(x^i)=v^i$ and $h_i(p)=\del_i|_pf$, so $Df=v^i\del_i|_pf$ as desired.<span style="float:right;">$\blacksquare$</span></span></div>

<div class="space"></div>
<p>  An immediate consequence is that $\dim T_pM=\dim M$, where the latter is the dimension of $M$ as a manifold. Moreover, this computation allows us to pull back the basis vectors $e_i\in\R^n$ to $\del_i|_\widehat{p}\in T_\widehat{p}\R^n$, which we can further pull back via the isomorphism $T_p\phi$ to a basis $\del_{x^i}|_p\coloneqq\frac{\del}{\del x^i}|_p\coloneqq(T_p\phi)^{-1}\del_i|_\widehat{p}$ for $T_pM$. This allows us to compute in $T_pf$ locally, which we do in the next section.</p>
<div class="space"></div>
<p><strong>Remark.</strong> Choosing<a href=https://zhaoshenzhai.github.io/mathwiki/vector_space.md/#existence_of_bases class="internalLink justifications dag" title="Vector Space" mathLink="" secID="existence_of_bases" secDisplay="Existence of Bases" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/vector_space.md/#existence_of_bases&#34;, {&#34;Date&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Vector Space&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/vector_space&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Vector Space&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/vector_space&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/vector_space.md/#existence_of_bases&#34;, {&#34;Date&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-30T19:27:52-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Vector Space&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/vector_space&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">$^\dagger$</a> a <a href=https://zhaoshenzhai.github.io/mathwiki/basis_module_theory.md class="internalLink references" title="Basis (Module Theory)" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/basis_module_theory.md&#34;, {&#34;Date&#34;:&#34;2024-08-31T09:09:40-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-31T09:09:40-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-31T09:09:40-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Basis (Module Theory)&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/basis_module_theory&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-08-31T09:09:40-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-31T09:09:40-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-31T09:09:40-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Basis (Module Theory)&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/basis_module_theory&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/basis_module_theory.md&#34;, {&#34;Date&#34;:&#34;2024-08-31T09:09:40-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-31T09:09:40-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-31T09:09:40-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Basis (Module Theory)&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/basis_module_theory&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">basis</a>, this computation gives a <a href=https://zhaoshenzhai.github.io/mathwiki/natural_transformation.md class="internalLink references" title="Natural Transformation" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/natural_transformation.md&#34;, {&#34;Date&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Natural Transformation&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/natural_transformation&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Natural Transformation&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/natural_transformation&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/natural_transformation.md&#34;, {&#34;Date&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Natural Transformation&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/natural_transformation&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">natural</a> isomorphism $T_pV\iso V$ for any $p\in V$ in a vector space.</p>
<h2 class="hide">Local Computations</h2>
<p>If $f:M\to N$ is smooth, choosing charts allows one to compute its tangent map $T_pf:T_pM\to T_{f(p)}N$. In fact, the matrix<a href=https://zhaoshenzhai.github.io/mathwiki/free_module.md/#homomorphisms_of_f.g._free_modules class="internalLink references dag" title="Free Module" mathLink="" secID="homomorphisms_of_f.g._free_modules" secDisplay="Homomorphisms of (f.g.) Free Modules" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/free_module.md/#homomorphisms_of_f.g._free_modules&#34;, {&#34;Date&#34;:&#34;2024-08-31T12:23:37-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-31T12:23:37-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-31T12:23:37-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Free Module&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/free_module&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-08-31T12:23:37-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-31T12:23:37-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-31T12:23:37-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Free Module&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/free_module&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/free_module.md/#homomorphisms_of_f.g._free_modules&#34;, {&#34;Date&#34;:&#34;2024-08-31T12:23:37-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-31T12:23:37-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-31T12:23:37-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Free Module&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/free_module&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">$^\dagger$</a> of $T_pf$ will be the <em>Jacobian</em> of the map $\psi\circ f\circ\phi^{-1}$, which generalizes the same thing from multivariable calculus:
$$\begin{equation}
T_pf\l(\l.\frac{\del}{\del x^i}\r|_p\r)=\l[\l.\del_i(\psi\circ f\circ\phi^{-1})^j\r|_\widehat{p}\r]\l.\frac{\del}{\del y^j}\r|_{f(p)}\eqqcolon J_\widehat{p}(\psi\circ f\circ\phi^{-1})\l.\frac{\del}{\del y^j}\r|_{f(p)}
\end{equation}$$</p>
<p>Finally, two charts $(U,\phi)$ and $(V,\psi)$ around $p$ induces two (possibly) different bases for $T_pM$, so we need to understand how they are related. This is done by applying the above to the map $\psi\circ\phi^{-1}$ and using functoriality of $T_p$ (i.e, the Chain Rule):
$$\begin{equation}
\l.\frac{\del}{\del x^i}\r|_p=\l(T_{\phi(p)}\phi^{-1}\r)(\del_i|_p)=\l(T_{\psi(p)}\psi^{-1}\r)\circ T_{\phi(p)}(\psi\circ\phi^{-1})(\del_i|_p)=\l(T_{\psi(p)}\psi^{-1}\r)\l(J_{\phi(p)}(\psi\circ\phi^{-1})\del_i|_p\r)=J_{\phi(p)}(\psi\circ\phi^{-1})\l.\frac{\del}{\del y^j}\r|_p.
\end{equation}$$
Moreover, if $v\in T_pM$ has basis representations $v^i\del_{x^i}|_p$ and $\widetilde{v}^j\del_{y^j}|_p$, then $\widetilde{v}^j=J_{\phi(p)}(\psi\circ\phi^{-1})v^i$ by matching components.</p>
<h2 id="velocity-vectors">Velocity Vectors</h2>
<p><span style="color:red"><strong>TODO</strong>.</span></p>
<h1 id="local-properties-of-smooth-maps">Local Properties of Smooth Maps</h1>
<p><span style="color:red"><strong>TODO</strong>.</span></p>
<h1 id="tangent-bundles-and-vector-fields">Tangent Bundles and Vector Fields</h1>
<p><span style="color:red"><strong>TODO</strong>.</span></p>


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                September 5, 2024 | Zhaoshen Zhai

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